Why is physics scarier than other math/science classes?

After some discussion with my student (geometry/chemistry) tonight, I think I have an answer to a question that plagued me for a while: why is physics scary?

It's scary because it's the first (and last) exposure to multi-step, multi-tool problem solving in math and science.
I had a high school student who found precalculus easy (or at least manageable, even that nasty stuff toward the end) but struggled mightily with physics problem solving. I couldn't figure it out for a while.

At first I thought it was that I found it easier to teach precalculus; the book is set up nicely, and I don't have to worry about abstract concepts (except insofar that I tried to tie it to useful stuff).

Then I thought maybe it had to do with the student having a weak science background. But by her own admission, she had both a weak math and science background.

It wasn't work ethic; this student worked a lot, both in class and outside. (I was dorm RA, so I knew studying was happening even outside of school hours.)

And I couldn't just chalk it up to "being Chinese", and the stereotype that Asian education consists of rote memorization and drills, leading to mathematical fluency but deficient creativity. (I secretly suspect this is an excuse perpetuated by Westerners to ignore the severe gap in educational readiness vis a vis other nations.) "Western" students in the same classes exhibited this pattern, too.

Eventually, it came down to this multi-step, multi-tool problem. Both she and I determined this independently.

***

In the typical high school math class, you solve problems by using a single trick or tool.

In biology, you mostly memorize a bunch of concepts and vocabulary, which are evidently important skills in the first year of med school.

Chemistry presents perhaps the strongest challenge to this thesis. It is possible to generate a multi-step, multi-tool problem in chemistry. There's a reaction, and you have to figure out it's yield and reaction type. First, you might have to do some stoichiometry to balance the equation. Then, you might have to figure out Lewis structures to determine the number and type of bonds, and then calculate the binding dissociation energies. You can then figure out if the equation is endothermic or exothermic. Maybe you adjust the calculation using phase transitions. Then you can determine molar masses, and compare the expected mass to the measured mass, or some other contrived number that allows you to calculate the yield.

So maybe chemistry is the first opportunity. But my experience indicates that plenty of students muddle through chemistry and hit a solid brick wall when they take physics. So there's something different about it.

I think chemistry, in principle, can be taught with emphasis on using multi-step, multi-tool problems to solve chemistry problems. But in practice, it looks like that's not done. I don't have a good explanation why that is; I've never taught a chemistry course. (Those who tutor, teach or study chemistry: your input is definitely welcome.)

***

In general, in the sciences, the problem statements are longer and more involved, making it less practical to have students do a number of problems on a single concept to hammer it in (as it's done in math). As much as we'd like to think otherwise, repetition and drills really do help cement a concept.

Maybe physics would be better if we could better segment subject material, and have more practice problems limited to one topic. This is different than how most textbooks are set up. Most of the better textbooks I've seen have, at the end of the chapter, problems grouped by topic, and prefaced with qualitative questions. It's not like a math book, where the section/chapter problems are divided into clusters in which you are asked to basically do the same thing over and over.

Physics, as a multi-step, multi-tool discipline, requires that all the tools work, and all the steps are clear. Maybe in other classes, even chemistry, a student can half-ass Lewis structures and still get an A. But it's just not possible to half-ass, say, linear momentum and be able to learn the rest. (Chemists: feel free to quibble and argue that the analogy ain't fair; I'll argue that even precious PV=nRT can be botched without irreparable damage to the rest of chemistry learning and the final grade.)

I'm not arguing that physics is better, or necessarily more complicated. But it is different.

Anyway, I think I'm going to revise how I teach physics. I basically need to generate drills, in addition to the problem-solving organizational methods that I'd ask them to use to convey their knowledge in a solution.

A math problem and a general life lesson

A former student highlighted a problem that I found too intriguing to resist. It was challenging, but I managed to solve it.

Let me be clear - the point of this post isn't to shore up my ego by demonstrating my ability to do algebra better than someone else. That is not the point at all. I think the student showed just the right judgment in struggling with it for a couple hours, then giving up and searching for an answer. That's not intellectually dishonest -- it's quite smart. I probably would've given up and not ask anyone -- which I did do, to a distressing level, in college.

The point is to show how something intimidating and new can be connected with prior knowledge. College, especially in technical disciplines, is largely about extending this prior knowledge in ways that are new, but connected with what you already know. Don't worry, you'll get practice, training, and experience, which will refine your judgment and improve your ability to solve problems of increasing complexity.

Here's the question:

Here's my handwritten solution. Sorry if the photo isn't great - my scanner is broken.

It took about an hour. Most of that time was spent on wrongheaded paths. Once I realized what would work, it took about ten minutes. By the way, that's probably an optimistic estimate of the amount of time solving a problem/knowing what to do versus the amount of time wandering like blind moles in research, in my own terrible experience.

 Some of my friends might find that ridiculously slow. I suppose I could say I'm rusty, but, to be honest, I never really put as much energy and thoughtfulness into mathematical derivation in school as, say, understanding psychology or international relations. It just never interested me -- or it frightened me.

Remember, the point of this post is to illustrate a point about math, and life. Sometimes, when confronted by something that seems impossible, it can help to look at hints for how to derive something.

Repeating, in case it's not clear from the scan.

In this case, there were two key insights:

1. The solution form looks like solving for tan (x/2) was the penultimate step. This suggested a substitution of a dummy variable equal to x/2.

2. The radical on one side of the equation in the penultimate step suggested a quadratic solution, with tan(x/2) as "x" in the quadratic form of the equation. From that, I was able to figure out what A, B, and C are. (Note that it doesn't matter whether A is positive or negative, but the relative sign of A, B and C are important.)

Based on that, I was able, after some time, to figure out the right approach and the right substitutions to make, and solved it quasi-formally.

Note:
In this case, looking at half-angle and double-angle formulas was briefly counterproductive. I saw a radical in the half-angle formulas, and a radical in the solution, so I thought that was the proper approach. Big mistake - I ended up wasting time just because of a superficial connection.

As with many things, looking for and identifying superficial connections between things can lead us astray.

For this problem, I needed to know how to recognize the solution of a quadratic equation, how to use a quadratic equation to solve for a function (versus a variable like x), how to substitute a variable for another one to take advantage of double-angle formulas, and a sense of which double-angle formulas I should use. (sin 2x is straightforward, while there are three choices for cos 2x; even though they are equivalent, not all are equally useful in a given situation. The handwritten work shows I momentarily chose the wrong one.)

So stuff from Algebra 2 could be put together to solve an algebraically intensive problem from first-year college physics.

Now I think this is generally true, even at the cutting edge of science, and perhaps other human endeavors. We build off of our prior knowledge. We extend our knowledge and tools to dazzling heights -- but, barring something really weird and advanced beyond my ken, every tool and every solution builds by analogy or by logical extension from previous ones.

So the next time you see something like this on a test -- don't panic! Take a deep breath. See if the form of the solution looks familiar. It might pay to make a list of the standard tricks -- in this particular case, substituting to take advantage of double-angle formulas, and solving for a function instead of a variable using the quadratic equation, with particular attention to coefficients.

Don't be afraid to work backwards -- sometimes it makes a lot more sense to do that. For the formal proof, you need to start from the beginning, but there's no rule saying you can't understand a derivation by reverse engineering it. Just practice, and try to see how anything that is "new" relates to things you know already. If you don't know those other, more fundamental things, make sure you do before you tackle the complicated things.

(Aside: I wish I thought this clearly 11 years ago. I would've probably done a bit better in college and gotten more out of it.)

Finally, to my former students -- sorry if sometimes it seemed like you were being thrown into the deep end of the pool when I was teaching. It was tough-- there were students at all different levels, and I think as an amateur teacher I didn't do as good a job as connecting the dots as I should've. Consider this a down payment on that deficit.

Now, left as an exercise to the reader: extending this analysis beyond math into daily life.

Calculus memories from Harvey Mudd College

I had a multivariable calculus professor in college who was first-generation Chinese. She's a world expert on differential geometry, and a bit quirky. How much of it is her being Chinese, and how much of it is her quirkiness, I don't know.

Evidently we weren't the brightest class in recent memory, because she decided to review some single-variable calc during a review session. Here's how she helped us remember the derivatives of the exponential and natural logarithmic functions.

"e^x is like a strong child. You hit it [with a derivative] and nothing happens."

"ln x is like a weak child. You hit it and it dies."

I think there was audible consternation. But we remembered.

WolframAlpha is interesting, but occasionally farts

I was using WolframAlpha recently as a graphing calculator. For those of you who don't know, WolframAlpha is a remarkable site that serves as a graphing calculator, computational tool, and aggregator of information. You can do a search for lots of things - not all technical in nature.

It is generally reliable, although I did run into a problem in which it returned only one of two correct roots (a quadratic involving sixth roots, or something like that; unfortunately, I didn't save the output.)

I was once curious about the demographic trends between Brians and Ryans in America. To my delight, it appears Ryan will soon become the dominant name in America.

Maybe Brians will have their names mangled, for a change.

But, occasionally, the outputs are really weird.

I'm not sure how I got this output - maybe I accidentally clicked on something. Maybe the book I was reading at the time somehow bumped the right set of keys. But I ended up with this.

Contrasts it with a query involving an actual celestial object.

I think God is telling me to get in touch with my roots. Either that, or Sanrio and WolframAlpha are engaged in a massive conspiracy to take over the United States using kawaii pilots.

70 years is still too soon. Who in the world thought this was a good idea? 

Why you should hug a math major today (and maybe even scientists and engineers)

I've been working on Slader, a website where you answer math questions and get paid a bit per solution. There's more comprehensive information in a review here, if you're interested.

Anyway, it occurred to me that the math majors/grad students I know are absolutely, totally nuts.

Look, I like LaTeX. It makes things pretty, especially equations. I've had to use quite a bit of it to write papers which, though scientifically moribund, were at least formatted (roughly) according to ApJ standards.

But you math people are totally ridiculous.

I remember one of my suitemates (let's call him "Jeff", because that's actually his name) had to write homework solutions in LaTeX. This was for an upper-division math class taught by a very assertive (though quite nice) Chinese professor. (Direct quote from the professor in my multivariable math class explaining derivatives: "e^x is like a strong child; you hit it and it stays the same. ln x is like a weak child; it dies and gets buried underground.)

So I'm sure he, and the other math majors I know, had to do LaTeX all the time.

Anyway, I just solved a part of a calculus problem. It involves determining where a function is concave up. It's part 1 of five. After completing it, I realized that it was totally not worth 75 cents.

Here's what the solution looks like:

Now wait a minute, you might say. That looks not bad. Well, it's actually quite bad, as in I did a crappy job. It's sparse on the explanations. I should add in subsections indicating that you need to test for the cases when x>0 and x<0. But I got tired. And here's why. Here's the LaTeX source code that I used to actually write the solution.

Now this isn't bad if you're a computer scientist, or a mathematician, or an engineer. But I taught high school for the last year. I have no fucking clue how people handle coding all day. Maybe it's like French; I don't know how the hell it works, but I guess if you do it long enough, it makes sense to you, even if it makes you seem mechanical (programming) or pretentious (French).

Anyway, I have a whole new sympathy for math people who do this every goddamn day, for many hours of said goddamn day. So go hug one. Maybe it will help with the carpal tunnel and eyestrain.

Sample question from a student, and my reply

Looking for feedback on whether I'm providing good help to my students via email. A sample (in fact, to this point, the only) email exchange for precalc.

How are you today? I am a student in your pre-calculus class.I have some questions on my homework.Could you please give me some help?

1.How to find out real zeros from all possible rational zeros of a function?(I have read the samples in the texrbook ,but I can't understand.Please give me a specific sample like page 160 :11)I know how to find all possible rational zeros ,but real zeros ,I can't.

2.Page 160:In exrcise 25-28 ,I have no idea to sketch the graph of f so that some of the possible zeros in part (a) can be disregarded)

Thank you for your help!Have a nice weekend.

My reply:

Hope you're enjoying your weekend so far!


1.How to find out real zeros from all possible rational zeros of a function?(I have read the samples in the texrbook ,but I can't understand.Please give me a specific sample like page 160 :11)I know how to find all possible rational zeros ,but real zeros ,I can't.

So the first thing to do is to use the rational zero test.

Remember that the rational zero test can be used on functions of the form

f(x) = an*x^n + a(n-1)*x^(n-1)+... + a1*x + a0.

where an is the coefficient of the term of order n (x^n).

We find the rational zeroes by taking the factors of the a0 term (the constant, which is -6) divided by the coefficient on the highest order term. In this case, the function is of order 3. The coefficient for x^3, a3, is 1.

If a0 is -6, the factors of a0 are {-6, -3, -2, -1, +1, +2, +3, +6}.

and the factors of a3 are {-1, 1}.

So, dividing the factors of a0 by the factors of a3 give us the possible rational zeroes:

a0/a3 = {-6, -3, -2, -1, 1, 2, 3, 6}.

I picked one root to try, +1. I substitute x = 1 into the function.

f(1) = (1)^3 - 6(1)^2 + 11(1) -6
= 0.

I got lucky! So I know that x = 1 is a rational root of f(x).

Now, there are two things I can do at this point.

1. Keep trying the other rational roots.

If you do this, you try the other possible rational roots (factors of a0 / factors of a3) and see if any of them are a zero.


2. Divide by x-1 to get a quadratic equation, which I will solve either by guessing or by using the quadratic formula.

Remember that if x = 1 is a root of f(x), (x - 1) is a factor of f(x), and we can divide f(x) by the factor to find the other roots.

This is the approach I used, because I didn't want to waste time in case there are no other rational roots. If x = 1 is the only rational root, then the quadratic formula will give us the real (or complex) roots, guranteed!

So, using long division,

(x^3 - 6x^2 + 11x - 6)/(x-1) = x^2 - 5x + 6


Now, I can guess the factors - it looks like x = 2 and x = 3 will work. Sure enough,

x^2 - 5x + 6 = (x - 2)*(x - 3)

which means the real roots of f(x) are x = {1, 2, 3}.

But it's a good idea to solve it using the quadratic formula, in case we can't factor it easily.

The quadratic formula can be used for any equation of the form

ax^2 + bx + c = 0

where a, b, and c are real numbers.

The solutions are found using this formula

roots = (- b +/- sqrt(b^2 - 4ac)/(2a)

where sqrt("something") equals the square root of "something".

For

x^2 - 5x + 6

a = 1, b = -5, and c = 6.

Using the formula,

roots = (- (-5) +/- sqrt((-5)^2 - 4*1*6))/(2*1)

= (5 +/- sqrt(25-24))/2

= (5 +/- 1)/2

= {2, 3}.

So we see that the solutions are x = 2 and x = 3.

In summary, for a polynomial function, you do the following steps:

1. Use the rational root test to find possible roots.

2. Try the roots and see if any of them work.

3. For every root that works, divide the function by its factor. For example, if the rational root is x = 4, you divide the function by the factor (x - 4).

4. After dividing, see if the function is of order 2 (looks like ax^2 + bx +c). If so, you can use the quadratic formula. This is how you get real roots that are irrational, and also complex roots.


2.Page 160:In exrcise 25-28 ,I have no idea to sketch the graph of f so that some of the possible zeros in part (a) can be disregarded)

The easiest way to graph it is to use your calculator. But if you don't have that, then the easiest way to graph is to find out what f(x) equals for a number of x values.

For example, on problem 25, f(x) = x^3 + x^2 - 4x - 4

Using the rational zero test, the possible rational roots are

x = {-4, -2, -1, 1, 2, 4}.

So I need to graph it at least over the domain of x:[-4, 4] (everywhere from x = -4 to x = 4).

I would write two columns like this

x f(x)

-4

-3

-2

-1

0

1

2

3

4

And calculate what f(x) is for each x.


x f(x)

-4 -36

-3 -10

-2 0

-1 0

0 -4

1 -6

2 0

3 20

4 60


If you draw the points and connect them, they look something like the picture I've attached in this email.

Note that from this picture, it's easy to eliminate -4, 1, and 4 as possible roots, but -2, 1, and 2 look like rational roots of f(x).

This is an easy example, but sometimes you'll get a function that is more complicated. That's why it's a good idea to graph it, especially in cases where you have a lot of possible rational roots to test.

Good luck! And thanks for asking good questions!

I got a job

I’ve just been hired by Excelsior school, a private boarding school in Pasadena, CA that serves about 60-70 predominately international students from East Asia and Russia. I will be teaching six subjects: pre-calculus, calculus, physics, chemistry, Algebra 2, and SAT math. I will be teaching five days a week, about six hours a day. Pay is $24/hr. Health benefits typically aren’t offered until one has worked there for a year, though I’ve got a verbal promise that I will be given them after one semester.

I don’t have a teaching credential. But, as the interviewer mentioned, he tends to view those credentials as secondary to skills involving classroom management, subject area knowledge, and organizational skills. He said, three times, over the job offer call that he has great confidence that I’ll be able to do the job. It might be something that is said to every new hire, but it made me feel better.

I expect this will be a pretty grueling job. I’m expected to take over from the current teacher within the course of about three days. At least they have textbooks and, I’ve been promised, clear guidelines from the principal about what are the objectives for each class over the course of a year.

I’m not completely sure what I’ve gotten myself into. But at least I’ll have a reason to be in Pasadena after school hours, and will tutor AP courses at another nearby location.

I’ll save celebration for when I’ve survived a few weeks. In the meantime, I will be soliciting everyone I know who is/was a physics/math/chem teacher for advice, materials, websites, and general psychological preparation. (Help me get the “Charge of the Light Brigade” out of my head whenever I think about this job.)

Learning some humility from a technical interview

I had a brutal interview today for a tutoring job. Writing it up as a lesson to myself and others about under-preparation and overconfidence going into an interview with a test component.

It started nicely enough –resume questions from the president, who is an econ major, and some jovial joking with a Caltech engineering major who had worked at JPL. I bedazzled with my complicated one-sentence statement of my previous research and my more understandable explanation of what “non-redundant aperture masking with adaptive optics” really meant. I displayed a comfort level with the subjects I’d be expected to tutor (biology, chemistry, physics, calculus, pre-calculus) and even tied in my behavioral econ knowledge to indicate how I can relate to individuals from different disciplines and career aspirations.

I was anticipating a diagnostic test, which I had prepared for by taking a sample Physics B AP test. I did reasonably well, missing a couple questions concerning induction and the lensmaker equation (apparently I forgot the sign convention for the focal length, where it is positive if the lens is convex and negative if it is concave).

Just as I’m about to be asked some qualitative physics questions from a high school physics textbook, another tutor arrives. He is apparently a Caltech senior physics major, currently applying to grad schools. His engineering colleague, seeing the physicist arrive, decides to have him ask me some questions.

How to calculate your 10,000th day on Earth

As usual, whenever I try to take a nap I think of something to keep me awake. This time, it was my 10,000th day of life.

It’s pretty straightforward to calculate, though one needs to exercise some caution when converting it into a memorable format (for example, October 15, 2010).

To convert into years, multiply the following factors

(Note: Throughout this post, I multiply conversion factors instead of simply dividing 10000 by 365 because experience has taught me that division without understanding units can be very, very confusing in the end.)

Whoopty doo. I'm sure you'll be able to impress girls at a bar with this knowledge.

However, it’s more fun to know the specific date of your 10,000th day of life on Earth. This will require us to calculate more carefully. I’ve deliberately used a lengthy approach to highlight the steps, and make sure it’s understood why each correction factor is applied.